12 research outputs found
Chabauty--Kim and the Section Conjecture for locally geometric sections
Let be a smooth projective curve of genus over a number field. A natural variant of Grothendieck's Section Conjecture postulates that every section of the fundamental exact sequence for which everywhere locally comes from a point of in fact globally comes from a point of . We show that satisfies this version of the Section Conjecture if it satisfies Kim's Conjecture for almost all choices of auxiliary prime , and give the appropriate generalisation to -integral points on hyperbolic curves. This gives a new "computational" strategy for proving instances of this variant of the Section Conjecture, which we carry out for the thrice-punctured line over
Refined Selmer equations for the thrice-punctured line in depth two
In [Kim05], Kim gave a new proof of Siegel's Theorem that there are only
finitely many -integral points on . One advantage of Kim's method is that it in
principle allows one to actually find these points, but the calculations grow
vastly more complicated as the size of increases. In this paper, we
implement a refinement of Kim's method to explicitly compute various examples
where has size which has been introduced in [BD19]. In so doing, we
exhibit new examples of a natural generalisation of a conjecture of Kim.Comment: 58 pages, comments welcom